#Exploration of the Kelly Wealth Process for AR(1)
#######################################################
# Illustrates sequential code development.
# This code is just to get you started.
# It "runs" but it definitely has some weird features that
# I will look at during the day today.
######################################################
#
# Version 1: Just to get a sense of the problem.
#########################################################

error.model=function(n){rnorm(n, sd=.01)}

# Note that errror.model is of class function. This is the class that is
# required in the assignment to the rand.gen argument of the arima.sim function.
 #More about this shortly.

# First, just the usual AR(1) code.

arreturns=arima.sim(model=list(ar=0.3), n=100,
n.start=200, start.innov=rnorm(200, sd=.01),
rand.gen=error.model )


#Now the wealth process

w=rep(1, 100)

for(j in 2:100){
w[j]=w[j-1]*( 1 + 0.3 *arreturns[j-1]*arreturns[j]/(0.09*arreturns[j-1]^2 +.0001) )
}

#Looking at the result.

w

# This is a wild ride. You can think of some issues. It would be useful to keep track
# of the bet sizes. If the bet fraction is more than 1, indicating leverage, then 
# serious code would either cap the max bet at 100% or else do the computation where
# we pay for our margin borrowing.

tsplot(w)

#########################################################

#Version 2.
# Let's wrap this stuff up in a function with FEWER embedded constants.
# We'll still keep the standard deviation of the model as an embedded constant.


KW=function(rho, num){

stderr=.01 
#Embedded constant
error.model=function(n){rnorm(n, sd=.01)}
#Not a great design, but it is the best I can do right now.

arreturns=arima.sim(model=list(ar=rho), n=num,
n.start=200, start.innov=rnorm(200, sd=stderr),
rand.gen=error.model )

w=rep(1, num)

for(j in 2:num){
w[j]=w[j-1]*( 1 + rho*arreturns[j-1]*arreturns[j]/(rho^2*arreturns[j-1]^2 +stderr^2) )
}

return(w)
}

#Testing...

w=KW(rho=.3, num=200)
tsplot(w)

#This works but the science is wild. We still have the issues of bet sizes bigger than one
#and no accounting for margin.

#It's also irritating that we needed to hard code the stadard deviation for the error model.
#Perhaps you will find out what the issue is here.

############################################################

# These two versions should be enough to get you going. 

# Here are three issues that are worth resolving.

# 1. Understanding the sizes of the bets. 
# It would be useful to print out the Kelly factions
# to help us understant the wildness of the wealth function.

# If bet factors are larger than one or smaller than -1 we should deal with the margin cost ---
# or do something.

# Working our the margin version of Kelly is really more than I expect people to do in 
# this assigmenent but it should be 
# understood as an issue.

# 2. Here it would be easy and appropriate to cap bet fractions to be in [-1,1]. 

# Why the minus? If the previous period return was negative, then we actually go short in this
# model.

# 3. S-Plus Mystery. Why can't we pass the model standard error as a value to the function KW? 

# I'll bet we can, but when I just followed my nose I got error messages. Perhaps someone in
# the class can give us a hint.